Week 10.2: Z-test & p-value

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# Week 10.2: Z-test & p-value > **Prerequisites:** Hypothesis Testing Intro ([Week 10.1: Introduction to Hypothesis Testing](/notes/01-foundation-bsma1004-stats-2-week10-25-hypothesis-testing-intro)), CLT ([Week 5.2: Law of Large Numbers & Central Limit Theorem](/notes/01-foundation-bsma1004-stats-2-week05-16-lln-cl...

Week 10.2: Z-test & p-value

Prerequisites: Hypothesis Testing Intro (Week 10.1: Introduction to Hypothesis Testing), CLT (Week 5.2: Law of Large Numbers & Central Limit Theorem) Cross-links: BSMA3012 (Linear Stat Models) — Wald tests Core question: How do we test hypotheses about the mean when variance is known?

1. Intuition: The Simplest Test

The Z-test is used when:

Testing $H_0: \mu = \mu_0$ vs $H_1: \mu \neq \mu_0$ (or one-sided)
Known population variance $\sigma^2$
Data is normal OR sample size is large (CLT) The test statistic:

Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1) \text{ under } H_0.

2. The p-value Approach

Definition (p-value) The p-value is the probability, assuming $H_0$ is true, of observing a test statistic as extreme or more extreme than the one actually observed. Decision rule: Reject $H_0$ if p-value $< \alpha$ . p-value for different alternatives:

Alternative	p-value
$H_1: \mu \neq \mu_0$	$2 \cdot \mathbb{P}(Z \geq \
$H_1: \mu > \mu_0$	$\mathbb{P}(Z \geq z_{\text{obs}})$
$H_1: \mu < \mu_0$	$\mathbb{P}(Z \leq z_{\text{obs}})$

3. Worked Examples

Example 1: Two-Sided Z-test

A manufacturer claims their batteries last 100 hours on average. Test this at

\alpha = 0.05

. Population SD is known to be 10 hours. Sample of 25 batteries gives

\bar{x} = 96

hours. Step 1:

H_0: \mu = 100

H_1: \mu \neq 100

. Step 2:

\alpha = 0.05

. Step 3:

Z = \frac{96 - 100}{10/\sqrt{25}} = \frac{-4}{2} = -2

. Step 4 (critical value):

z_{0.025} = 1.96

. Reject if

|Z| > 1.96

. Step 5:

|Z| = 2 > 1.96

. Reject $H_0$ . Step 6 (p-value):

2 \cdot \mathbb{P}(Z \geq 2) = 2 \cdot 0.0228 = 0.0456 < 0.05

. Conclusion: There is sufficient evidence that the mean battery life differs from 100 hours.

Example 2: One-Sided Z-test

Test if student scores have increased from last year's mean of 70. Known SD = 15. Sample of 36 students:

\bar{x} = 74

\alpha = 0.05

. Step 1:

H_0: \mu = 70

H_1: \mu > 70

. Step 2:

\alpha = 0.05

. Step 3:

Z = \frac{74 - 70}{15/\sqrt{36}} = \frac{4}{2.5} = 1.6

. Step 4: Critical value

z_{0.05} = 1.645

. Reject if

Z > 1.645

. Step 5:

1.6 < 1.645

. Fail to reject $H_0$ . Step 6 (p-value):

\mathbb{P}(Z \geq 1.6) = 0.0548 > 0.05

. Conclusion: Insufficient evidence that mean has increased.

4. Critical Values vs p-value

(Diagram) Both approaches give the same decision.

5. Practice Questions

Q1 (Easy)

n=64

\bar{x}=52

\sigma=8

. Test

H_0: \mu=50

H_1: \mu \neq 50

\alpha=0.05

Full Solution

$Z = (52-50)/(8/\sqrt{64}) = 2/(1) = 2$ . $z_{0.025} = 1.96$ . $|Z| = 2 > 1.96$ → Reject $H_0$ . p-value $= 2(0.0228) = 0.0456 < 0.05$ .

Q2 (Medium)

n=100

\bar{x}=980

\sigma=50

. Test

H_0: \mu=1000

H_1: \mu < 1000

\alpha=0.01

Full Solution

$Z = (980-1000)/(50/10) = -20/5 = -4$ . $z_{0.01} = -2.326$ . $Z = -4 < -2.326$ → Reject $H_0$ . p-value $= \mathbb{P}(Z < -4) \approx 0.00003 < 0.01$ .

Q3 (Hard)

A factory claims defect rate ≤ 1%. Sample of 500 items finds 8 defects. Test at

\alpha = 0.05

Full Solution

$H_0: p = 0.01$ , $H_1: p > 0.01$ . $\hat{p} = 8/500 = 0.016$ .

$Z = \frac{0.016 - 0.01}{\sqrt{0.01 \cdot 0.99 / 500}} = \frac{0.006}{\sqrt{0.0000198}} = \frac{0.006}{0.00445} \approx 1.35$ .

$z_{0.05} = 1.645$ . $1.35 < 1.645$ → Fail to reject $H_0$ . Insufficient evidence that defect rate exceeds 1%.

Next topic: Week 11.1: t-test & Chi-Square Test — t-test and Chi-square test. Join Discord PreviousWeek 10.1: Introduction to Hypothesis TestingNextWeek 11.1: t-test & Chi-Square Test

What this note is really saying

Week 10.2: Z-test & p-value

1. Intuition: The Simplest Test

2. The p-value Approach

3. Worked Examples

Example 1: Two-Sided Z-test

Example 2: One-Sided Z-test

4. Critical Values vs p-value

5. Practice Questions

Q1 (Easy)

Q2 (Medium)

Q3 (Hard)

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